Probit Versus Logit Models

Ordinary linear regression predicts an unbounded number, but plenty of questions a quant faces are yes/no: will this bond default, will this trade fill, will this IPO price above range? Fit a straight line to a 0/1 outcome and it will happily predict probabilities of 1.4 or −0.2, which are meaningless. Probit and logit models fix this by squashing a linear score through an S-shaped curve that's pinned between 0 and 1 no matter how extreme the score gets — they just pick two very slightly different S-shapes to do it.

An analogy: two rulers that read almost the same

Imagine two rulers for measuring the same thing, one marked in a slightly different unit than the other but both spanning the same physical length. Convert between them and, for most everyday measurements, the numbers you get are close enough not to matter — the underlying quantity being measured is identical, only the labeling convention differs. Probit and logit are like this: both squash a linear score into a probability between 0 and 1, both fit by the same method, and for most datasets their fitted probabilities track each other almost exactly across the middle of the range. The real choice between them is about convenience of interpretation, not about which is "more correct."

The models, one piece at a time

Both start from the same linear score, $z_i = \beta_0 + \beta_1 x_{1i} + \cdots$, built exactly like an ordinary regression. The two models differ only in the function $F$ used to turn $z$ into a probability:

$$P(y_i = 1 \mid x_i) = F(z_i), \qquad F_{\text{logit}}(z) = \frac{1}{1+e^{-z}}, \qquad F_{\text{probit}}(z) = \Phi(z),$$

where $\Phi$ is the cumulative distribution function of the standard normal distribution — the probability that a standard normal random variable falls below $z$. In plain English: logit assumes the "noise" pushing an observation across the yes/no threshold follows a logistic distribution, while probit assumes that noise is normally distributed; both are symmetric, bell-shaped, S-curve-producing choices, and the logistic curve only has slightly fatter tails than the normal one. Both are fit by maximum likelihood, choosing $\beta$ to make the observed 0/1 outcomes as probable as possible under the model.

The practical difference is in the coefficients' scale, not their implied probabilities: logit coefficients are commonly reported as odds ratios ($e^{\beta_1}$, how much the odds of the outcome multiply per unit of $x_1$), which has no clean probit equivalent — probit coefficients don't have a similarly popular closed-form interpretation and are usually reported as marginal effects on probability instead.

Worked example 1: predicting a bond default

A credit model scores a bond's default probability from its leverage ratio $x$, fit by logit as $z = -3 + 1.5x$. At $x=1$ (leverage ratio of 1), $z = -1.5$, and $F_{\text{logit}}(-1.5) = 1/(1+e^{1.5}) \approx 0.182$ — about an 18.2% predicted default probability. A probit fit on the same data typically comes out close to $z = -1.9 + 0.95x$ (probit coefficients are usually smaller in magnitude, roughly by a factor of 1.6-1.7, because $\Phi$ is steeper near the middle than the logistic function); at $x=1$, $z=-0.95$, and $\Phi(-0.95) \approx 0.171$ — an 17.1% predicted probability, close to logit's 18.2%.

Worked example 2: odds ratio versus marginal effect

Suppose the logit fit above gives $\beta_1 = 1.5$ on leverage. The odds ratio is $e^{1.5} \approx 4.48$: each one-unit increase in leverage ratio multiplies the odds of default by about 4.48. This is a clean, scale-free statement. The equivalent probit marginal effect (the change in default probability, not odds, for a small change in $x$ near a given point) must instead be computed as $\phi(z) \times \beta_1$ where $\phi$ is the normal density — at $z=-0.95$, $\phi(-0.95)\approx 0.254$, giving a marginal effect of about $0.254 \times 0.95 \approx 0.241$, meaning roughly a 24-percentage-point increase in default probability per unit of leverage at that point on the curve — a number that itself changes at different points on the S-curve, unlike the odds ratio, which is constant everywhere.

Drag the mean and standard deviation to see the normal density $\phi(z)$ that underlies the probit link — it's the slope of $\Phi(z)$ at each point, which is why probit's marginal effects vary across the curve while logit's odds ratios don't.

What this means in practice

Logit dominates in practice mainly because odds ratios are easy to communicate and logit is what most binary classifiers and credit-scoring models default to, but probit shows up wherever the modeling story is explicitly about an underlying normal latent variable — most notably as the first stage of a Heckman selection model (see The Heckman Selection Correction). For a quant deciding between them, the fitted probabilities and classification performance are usually nearly indistinguishable; the choice is about which coefficient interpretation the audience wants.